# Hydrogen Matrix Lab 0.1.0 — checked result

## Concrete result

The exact idealized hydrogen `1s` density was evaluated on a `129 x 129 x 129` Cartesian matrix over `[-8 a0, +8 a0]` in each direction. A static mesh was then built at the matrix-derived radius enclosing 90% of the represented probability.

| Quantity | Finite matrix | Exact continuous value |
|---|---:|---:|
| Integrated probability | 1.000038674597 | 1 |
| Mean radius | 1.499896822 `a0` | 1.5 `a0` |
| RMS radius | 1.731952244 `a0` | sqrt(3) `a0` |
| 50% mass radius | 1.334634782 `a0` | 1.337030157 `a0` |
| 90% mass radius | 2.660474582 `a0` | 2.661160169 `a0` |
| 99% mass radius | 4.201934078 `a0` | 4.202973457 `a0` |

Using the NIST 2022 CODATA value `a0 = 5.29177210544e-11 m`, the matrix-derived 90% envelope radius is **140.7863 pm**. Its absolute radial error from the exact continuous result is **0.0006856 a0**, or about **0.0363 pm**.

The exported 90% surface contains 8,320 vertices and 16,128 triangles. Because the `1s` density depends only on radius, this surface is spherical. That symmetry is an output of the model and matrix, not evidence that an electron occupies a solid sphere.

## Independent verification

A separate script that does not import the implementation recomputed the radial normalization and moments using SciPy quadrature and recomputed the voxel integrals directly from the saved NPZ matrix. All seven declared checks passed:

- continuous normalization, mean radius, and mean-square radius;
- matrix normalization, mean radius, and RMS radius;
- equality of the three matrix second moments to numerical precision.

The five unit-test groups also passed.

## Status and limitations

- **Written mathematics:** the non-relativistic point-proton Schrödinger `1s` wavefunction and its analytic radial integrals.
- **Finite computation:** voxel integration, probability radii, sampled cloud, PNG preview, and OBJ mesh.
- **Measured evidence:** none was collected in this lab.
- **Hypothesis/theory status:** this result validates the implementation against standard hydrogen quantum mechanics; it does not validate a new ontology, a literal electron path, seven physical dimensions, P=NP, or a theological identification.

The model omits proton recoil, spin, relativity, fine and hyperfine structure, the Lamb shift, finite proton size, external fields, and environmental interactions. The proton marker in the visual is deliberately enlarged.

## Next bounded test

Add the `2p_z` state, verify its nodal plane and normalization independently, and compare its two-lobed signed wavefunction with its nonnegative probability density. That will test whether the matrix engine can represent genuine orbital structure rather than only a radially symmetric case.